GSP Investigation: Points of Concurrency
Medians in a Triangle
A median in a triangle connects a vertex with the midpoint of the opposite side.
1. Construct triangle ABC.
To construct a midpoint,
highlight the line segment,
go to Construct, and
choose Midpoint.
2. Construct the midpoints of the three sides.
3. Construct two of the three medians, each
connecting a vertex with the midpoint of its
opposite side.
4. Construct the intersection of the two
medians.
5. Construct the third median.*
Question 1: What do you notice about this third median? Drag a vertex of the triangle to confirm that this conjecture
holds for any triangle.
To label a point, click on
the Text tool (“A”) and
double-click on the point.
6. The point where the medians intersect is
called the centroid. Label the point Ce for
centroid.
7. Measure the distance from B to Ce and the
distance from Ce to the midpoint F.
8. Drag the vertices of ∆ABC and look for a
relationship between BCe and CeF.
To make a table, select
the two measurements.
Then, in the Number
menu, choose Tabulate.
9. Make a table with these two measures.
10. Change the triangle and double click on
the table values to add another entry. Keep
changing the triangle and adding entries to
your table.*
Question 2: What relationship do you see between the distances BCe and CeF?
Perpendicular Bisectors in a Triangle
In this investigation, you’ll discover properties of perpendicular bisectors in a triangle. You’ll also learn how to construct
a circle that passes through each vertex of a triangle.
1. Construct triangle ABC.
2. Construct the midpoints of the sides.
Select a side and its
midpoint. Then, in the
Construct menu, choose
Perpendicular Line.
3. Construct two of the three perpendicular
bisectors in the triangle.
4. Construct point G, the point of intersection
of these lines.
5. Construct the third perpendicular bisector.
Question 1: What do you notice about this third perpendicular bisector? Drag a vertex of the triangle to confirm that
this conjecture holds for any triangle.
6. Drag a vertex around so that point G moves into and out of the triangle. Observe the angles s
of the triangle as you do this.
Question 2: In what type of triangle is point G outside the triangle? Inside the triangle?
Select point G and a
vertex. Then, in the
Measure menu, choose
Distance. Repeat for the
other two vertices
7. Measure the distances from point G to each of the three vertices.
8. Drag a vertex of the triangle and observe the distances.
Question 3: The point of intersection of the three perpendicular bisectors is called the circumcenter of the triangle.
What do you notice about the distances from the circumcenter to the three vertices of the triangle?
9. Construct a circle with center G and radius
endpoint A. This is the circumscribed circle of
∆ABC.*
Altitudes in a Triangle
In this investigation you’ll discover some properties of altitudes in a triangle. An altitude is a perpendicular segment
from a vertex of a triangle to the opposite side. The side where the altitude ends is the base for that altitude and the
length of the altitude is also the height of the triangle from that base. Because a triangle has three sides, it also has
three altitudes.
1. Construct triangle ABC.
2. Construct a line perpendicular to AC
through point B.
Question 1: As long as your triangle is acute, this perpendicular line should intersect a side of the triangle. Drag point B
so that the line falls outside the triangle. Now what kind of triangle is it?
Select both endpoints of
segment AC, go to
Construct, and choose
Line.
3. With the perpendicular line outside the
triangle, use a line to extend side AC so that it
intersects the perpendicular line.
4. Construct point D, the point of intersection
of the extended side and the perpendicular
line.
5. Hide the two lines.
̅̅̅̅. Segment BD is an altitude.
6. Construct 𝐵𝐷
7. Drag vertices of the triangle and observe
how your altitude behaves.
Question 2: Where is your altitude when ∠A is a right angle?
8. Drag your triangle so that it is acute again (with the altitude falling inside the triangle).
9. Construct altitudes from all the triangle’s vertices.
10. Construct their point of intersection. This point is called the orthocenter of the triangle.*
Angle Bisectors in a Triangle
In this investigation you’ll discover some properties of angle bisectors in a triangle.
1. Construct triangle ABC.
Select the two segments
and the vertex that make
up the angle. Then in the
Construct menu, choose
Angle Bisector.
2. Construct the bisectors of two of the three
angles.
3. Construct point D, the point of intersection
of the two angle bisectors.
4. Construct the bisector of ∠C.*
Question 1: What do you notice about this third angle bisector? Drag each vertex of the triangle to confirm that this
observation holds for any triangle.
Select point D and one
side of the triangle. Then
in the Measure menu,
choose Distance. Repeat
for the other two sides.
5. Measure the distance from D to each of the three sides.
6. Drag each vertex of the triangle and observe the distances.
Question 2: The point of intersection of the angle bisectors in a triangle is called the incenter. What do you notice about
the distance from the incenter of a triangle to the three sides?
7. An inscribed circle is a circle inside a triangle that touches each of the three sides at one point
(in other words, it is a circle that is tangent to all three sides). Construct an inscribed circle that
stays inscribed no matter how you drag the triangle. (Hint: you’ll need to construct the radius of
the triangle first.)*